Uniform Nekhoroshev estimates on quantum normal forms

Abstract

We consider a system of completely resonant harmonic oscillators with a bounded analytic perturbation admitting an LI Fourier transform. We prove that the corresponding Schrodinger operator admits a normal form satisfying a uniform Nekhoroshev estimate: namely the remainder is exponentially small in the perturbation parameter uniformly in the classical limit. In the particular case of a single degree of freedom the above result yields a quantization formula which holds up to exponentially small terms.